# Carnots Theorem

## Introduction

The first law of thermodynamics states that in all closed systems, the total energy of the system is constant—it cannot be created or destroyed. This means that it’s impossible to make heat without also making work, and vice versa. The second law states that the entropy of an isolated system can never decrease, but must increase over time as it approaches equilibrium with its surroundings. However, there are ways to optimize thermal efficiency—one of these is Carnot’s theorem.

## Overview of Carnot's Theorem

In a thermodynamic system, heat cannot be converted completely into work. This is due to a concept known as thermal efficiency. Carnot's Theorem is an important result in thermodynamics that describes how much work can be extracted from a heat engine operating between two temperatures.

The author was actually Nicolas Léonard Sadi Carnot, and his paper paved way for Thermodynamics to become a science. One of his first discoveries described how heat energy could be converted into mechanical energy and can be used to do work such as powering an engine. This process is what drives steam engines; it is also known as heat-engine theory or simply, Carnot's theorem.

### What is a Reversible and Irreversible Engine?

To fully understand how and why heat engines work, we need to define what is meant by a reversible and irreversible engine. A reversible engine is one where for every step of input energy it only produces as much output energy. An irreversible engine on the other hand converts part of its input energy into useless heat and therefore can never be fully efficient. In fact, if an irreversible engine were 100% efficient than any temperature difference would produce infinitely large work out of it, something that could not be happening in reality.

This means that all real-world engines are irreversibly losing some of their energy so they will always have some level of efficiency less than 1. However, there are ways to make them more efficient than others; for example, steam turbines or jet engines are around 50% efficient while combustion engines such as those found in cars can have efficiencies up to 80%. So now we know what these terms mean let’s look at Carnot’s theorem itself.

## Statement of the Carnot’s Theorem

Engines that have hot and cold ends between the same two reservoirs, the Carnot engine, are more efficient than engines with the same hot and cold ends but which are working between different reservoirs. This can be proved by using a reversible process. The aim of this note is to prove and explain Carnot’s theorem from the first principles.

It doesn't matter how the operation occurs; all Carnot engines are equally efficient between two heat reservoirs. It means that if there are two heat reservoirs at temperatures T1 and T2, then a Carnot engine operating between them will be as efficient as it can possibly be. As long as we operate our engine on those temperature levels, we cannot improve its efficiency. If you’re not sure what I mean by efficiency in the context of engines, think about work done divided by energy consumed.

Images Coming soon

Let us consider two heat reservoirs at temperatures T1 and T2(T1>T2), R is the reversible engine and I is an irreversible engine. A denotes Irreversible and B is reversible.

For a reversible engine, Q1 is the heat taken by engine 1, and reject heat is Q2. Work done W1 is,

$$\mathrm{W_1\:=\:Q_1\:-\:Q_2}$$

For an Irreversible engine,

$$\mathrm{W_1'\:=\:Q_1'\:-\:Q_2'}$$

Common Source of Heat is,

$$\mathrm{Q_1A\:=\:Q_2B}$$

Efficiency is:

$\mathrm{\eta_A\:=\:W_A/Q_1A|\eta_B\:=\:W_B/Q_1B}$

If $\mathrm{\eta_A \gt \eta_B\:=\:\gt\:W_A\:\gt\:W_B}$

$\mathrm{\Rightarrow\:W_A-W_B\:\gt\:0}$ (Equation 1)

The heat engine is transformed into a heat pump after the direction of the work is reversed. In this way, the work output of A becomes a work input for B, which functions as a heat pump.

Work Output is:

$\mathrm{W_A-W_B\:\gt\:0}$, We get Net positive work output.

$$\mathrm{\eta_A\:\gt\:\eta_B\:or\:\eta_{Reversible}\gt \eta_{Irreversible}}$$

## Applications of Carnot’s Theorem

Here are some applications and limitations of Carnot’s theorem:

• A heat engine is an example of a thermodynamic system.

• A refrigerator is an example of a heat pump.

• Thermodynamic systems can be used to do work.

• Heat flows naturally from hot objects to cold ones.

• There are no perpetual motion machines in our universe.

• We can use Carnot's theorem to calculate how much work can be done by a heat engine.

• We can use Carnot's theorem to calculate how much energy is needed to cool something down

## Limitations of Carnot’s Theorem

There are four major limitations of Carnot’s theorem as follows:

• It describes a cycle without friction. That is why an ideal engine must have zero friction.

• It does not describe a cycle that starts at temperature T1 and ends at temperature T2, but rather one that starts at a higher temperature and goes to a lower temperature.

• It describes a reversible process in which there is no entropy change; however, many processes are irreversible in which entropy increases.

• Finally, there is no mention of heat flow into or out of the system; thus we do not know whether heat flows into or out of an engine during operation.

## FAQs

Q1. What is the expression used to find efficiency obtained from the Carnot engine?

Ans. $\mathrm{Η_{max} \:= \:T_H \:- \:T_C / T_H}$

The mechanical efficiency of an engine is known as work done per unit of heat supplied.

Q2. Who gave the Carnot engine?

Q3. What is the COP for a reversible refrigerator?

Ans. One of these is a refrigerator, which cools its contents by transferring heat from an external source to itself. Since no heat is transferred to or from its surroundings, it must have zero COP.

Q4. What is the efficiency and work ratio of a simple gas turbine cycle?

Ans. The efficiency ratio is extremely low compared to the work ratio.

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Updated on: 13-Oct-2022

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